Base field 4.4.18097.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 6x + 4\); narrow class number \(2\) and class number \(1\).
Form
Weight: | $[2, 2, 2, 2]$ |
Level: | $[16, 4, -w^{3} + 6w]$ |
Dimension: | $12$ |
CM: | no |
Base change: | no |
Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{12} - 19x^{10} + 122x^{8} - 339x^{6} + 427x^{4} - 212x^{2} + 16\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
3 | $[3, 3, -w + 1]$ | $\phantom{-}e$ |
4 | $[4, 2, w]$ | $\phantom{-}0$ |
4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $-\frac{3}{4}e^{11} + \frac{107}{8}e^{9} - 76e^{7} + \frac{335}{2}e^{5} - \frac{1087}{8}e^{3} + 23e$ |
7 | $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ | $-\frac{21}{8}e^{10} + \frac{93}{2}e^{8} - \frac{1041}{4}e^{6} + \frac{4417}{8}e^{4} - 399e^{2} + 30$ |
13 | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $\phantom{-}\frac{43}{8}e^{10} - 95e^{8} + \frac{2119}{4}e^{6} - \frac{8951}{8}e^{4} + \frac{1615}{2}e^{2} - 64$ |
17 | $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ | $\phantom{-}e^{11} - 18e^{9} + 104e^{7} - 235e^{5} + 193e^{3} - 29e$ |
27 | $[27, 3, -w^{3} + 7w + 1]$ | $-\frac{3}{8}e^{11} + \frac{13}{2}e^{9} - \frac{139}{4}e^{7} + \frac{527}{8}e^{5} - 29e^{3} - 19e$ |
31 | $[31, 31, w + 3]$ | $-\frac{27}{4}e^{10} + \frac{239}{2}e^{8} - \frac{1337}{2}e^{6} + \frac{5683}{4}e^{4} - \frac{2083}{2}e^{2} + 92$ |
31 | $[31, 31, -w^{2} + 5]$ | $\phantom{-}\frac{9}{4}e^{11} - \frac{159}{4}e^{9} + \frac{443}{2}e^{7} - \frac{1871}{4}e^{5} + \frac{1363}{4}e^{3} - 31e$ |
37 | $[37, 37, w^{2} - 3]$ | $\phantom{-}\frac{11}{2}e^{11} - \frac{389}{4}e^{9} + 543e^{7} - 1152e^{5} + \frac{3385}{4}e^{3} - 78e$ |
41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ | $-\frac{19}{4}e^{11} + \frac{335}{4}e^{9} - \frac{929}{2}e^{7} + \frac{3873}{4}e^{5} - \frac{2691}{4}e^{3} + 33e$ |
47 | $[47, 47, w^{3} - 5w - 3]$ | $\phantom{-}\frac{11}{8}e^{11} - 24e^{9} + \frac{523}{4}e^{7} - \frac{2127}{8}e^{5} + \frac{375}{2}e^{3} - 25e$ |
53 | $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ | $-\frac{49}{8}e^{11} + \frac{431}{4}e^{9} - \frac{2381}{4}e^{7} + \frac{9865}{8}e^{5} - \frac{3397}{4}e^{3} + 40e$ |
61 | $[61, 61, w^{3} - w^{2} - 5w + 3]$ | $-\frac{57}{8}e^{10} + \frac{251}{2}e^{8} - \frac{2781}{4}e^{6} + \frac{11605}{8}e^{4} - 1025e^{2} + 72$ |
83 | $[83, 83, w^{3} + w^{2} - 6w - 7]$ | $\phantom{-}\frac{3}{8}e^{11} - \frac{27}{4}e^{9} + \frac{155}{4}e^{7} - \frac{675}{8}e^{5} + \frac{229}{4}e^{3} + 7e$ |
83 | $[83, 83, -w^{3} + 5w + 1]$ | $\phantom{-}\frac{7}{4}e^{10} - \frac{63}{2}e^{8} + \frac{363}{2}e^{6} - \frac{1619}{4}e^{4} + \frac{641}{2}e^{2} - 44$ |
83 | $[83, 83, 2w - 1]$ | $-\frac{1}{4}e^{11} + \frac{17}{4}e^{9} - \frac{43}{2}e^{7} + \frac{131}{4}e^{5} + \frac{43}{4}e^{3} - 43e$ |
83 | $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ | $\phantom{-}\frac{61}{8}e^{10} - 135e^{8} + \frac{3021}{4}e^{6} - \frac{12833}{8}e^{4} + \frac{2335}{2}e^{2} - 98$ |
89 | $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ | $-\frac{1}{2}e^{8} + 8e^{6} - 37e^{4} + \frac{109}{2}e^{2} - 18$ |
89 | $[89, 89, w^{3} - 5w + 5]$ | $\phantom{-}\frac{5}{4}e^{10} - \frac{45}{2}e^{8} + \frac{259}{2}e^{6} - \frac{1145}{4}e^{4} + \frac{425}{2}e^{2} - 18$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$4$ | $[4, 2, w]$ | $-1$ |